No group literally starts that way, but their behaviors exhibit as such and oftener there is a reason why terms like Oppression Olympics exist in the social media lexicon. It's not out of humor but that there's a core truth to it.
For example, there's a segment of gender critical feminists that believe that transgender activists are uniformly assaulting them on all fronts when in reality transgender activists have been focused on issues of non-discrimination in law (public accommodation, employment, and healthcare as big issues where activists seek redress). Neither side assumes one is special or better/worse than the other but both attempt grand standing on who is more oppressed/harmed by society. And how unique their position is in terms of said oppression. I've rarely seen either side (being trans myself here) extend a hand in regards to trying to understand the other. It seems the most vocal members of this fracas are doomed to bury each other in terrible memes and virtual shouting matches. Some have gone further like Cathy Brennan who has doxxed a transgender teenager (who had no quarrel with her or other gender critical feminists) for the Pacific Justice Institute (a right wing think tank out of California). So, I hope you can see where I'm coming from here.